"Probabilistic Chess Ratings" (Odds Ratings) is a 24 page concise theory in PDF (please eMail me at [email protected]). A rating difference Ro is transformed using the Rating Difference Transformation Function (RDTF) R: R(K, Ro, S) = (1/K).ln{[e^(K.Ro.{Ro > 0}) + S] / [e^(-K.Ro.{Ro < 0}) + 1 - S]} The inequalities are 1 or 0 as usual, S is the Success 0 ≤ S ≤ 1 continuous in other applications where measurable, discrete S in {0, ½, 1} for chess. This implies that draws in chess will produce no change if Ro = 0, otherwise correction is scaled in the appropriate direction. The system scaling constant K produces our prefered range and is derived using: K = -(1 / Rs).log[(1 / Ps) – 1] Where: Ps = system probability constant; Rs = system rating difference constant; Rp = provisional rating. Rp = 1500, Ps = 2/3 and Rs = 100 seem ideal and allow the easiest mental odds estimates. With odds as Wins Per Loss (WPL): R = 100 implies 2 WPL R = 200 implies 4 WPL R = 300 implies 8 WPL R = 400 implies 16 WPL ... R = 1,000 implies 1,024 WPL ... R = 2,000 implies 1,048,576 WPL ... Convergence to the manifesting probabilities (see PDF) is facilitated by "The Fundumental Theorem of Games of Skill" (FTGS). Given contestants A, B and C: A scores m WPL against B and B scores n WPL against C implies that A will score m x n WPL against C Equivalently for probabilities: If P(AB) = p, P(BC) = q, and P(AC) = r then r = p & q = pq / [pq + (1 - p)(1 - q)] For rational arguments, we have rational results, which may be concatenated mentally! a/c & b/d = a.b / [a.b + (c - a).(d - b)] The binary concatenation operator "&" forms an Abelian group over probabilities with identity 1/2 and inverse 1 - p. This simplifies some important derivations in the complete theory, which analyses the significance of wins and draws in the struggle for rating points. The axiom reflects the reality that probabilities converge to the asymptotes p = 0 and p = 1 exponentially with increasing rating difference. It may be that "Games of Skill" (0 < P < 1) differ only in the probability represented by one standard deviation, a measure of the "masterability" that a species (orangutan, man, machine, &c.) may acquire. This conjecture is statistically contestable. R = R0 in RDTF if and only if the value of S is equal to the probability represented by the rating difference R0. The hyperbolic tangent curve (Rating Difference vs Probability, scaled & translated): P = (1/2) + (1/2).tanh[(1/2).K.R] is the axiomatic basis from which FTGS and ultimately RDTF are derived. In terms of Probabilities: P = 1 / [1 + e^(-K.R)] The alternative WPL formula is trivial, clearly exponential, and left as an exercise. In practice, rating changes MUST be calculated using RDTF chronologically, round per round, match per match, with pairings in tournaments minimising the total WPL each round as tournament rules permit. ©